In this study, we prove an existence result of renormalized solutions for nonlinear parabolic equations of the type $$ displaystylefrac{partial b(x,u)}{partial t} -mbox{div}>a(x,t,u,nabla u)-mbox{div}> Phi(x,t,u)= f quadmbox{in }{Q_T=Omegatimes (0,T)}, $$ where $b(x,cdot)$ is a strictly increasing $C^1$-function for every $xinOmega$ with $b(x,0)=0$, the lower order term $Phi$ satisfies a natural growth condition described by the appropriate Orlicz function $M$ and $f$ is an element of $L^1(Q_T)$. We don't assume any restriction neither on $M$ nor on its conjugate $overline{M}$.
We introduce the concept of Isbell convexity in fuzzy quasi-metric spaces, which we call fuzzy Isbell convexity. This idea extends Isbell convexity (or q-hyperconvexity) in quasi-metric spaces to fuzzy quasi metric spaces. We show that fuzzy Isbell convexity is preserved by certain $F$-bounded subsets and the space of non-negative function pairs of the fuzzy quasi-metric space.
In this paper, for admissible and integrable function $psi$ in $L^2(mathbb{R}^n)$, the multi-dimensional continuous wavelet transform on Sobolev spaces is defined. The inversion formula for this transform on Sobolev spaces is established and as a result it is concluded that there is an isometry of Sobolev spaces $H_s(mathbb{R}^n)$ into $H_{0,s}(mathbb{R}^n times mathbb{R}^+_0times S^{n-1})$, for arbitrary real $s$. Also, among other things, it is shown that the range of this transform is a reproducing kernel Hilbert space and the reproducing kernel is found.
Let $(X,d)$ and $(Y,rho)$ be compact metric spaces, $tau$ and $eta$ be Lipschitz involutions on $ X$ and $Y$, respectively, $mathcal{A}=Lip(X,d,tau)$ and $mathcal{B}=Lip(Y,rho,eta)$, where $Lip(X,d,tau)=lbrace fin Lip(X,d):fcirctau=bar{f}rbrace $. For each $fin mathcal{A}$, $sigma_{pi,mathcal{A}}(f)$ denotes the peripheral spectrum of $f$. We prove that if $S_{1},S_{2}:mathcal{A}rightarrow mathcal{A}$ and $T_{1},T_{2}:mathcal{A}rightarrow mathcal{B}$ are surjective mappings that satisfy $sigma_{pi,mathcal{B}}(T_{1}(f)T_{2}(g))=sigma_{pi,mathcal{A}}(S_{1}(f)S_{2}(g))$ for all $f,gin mathcal{A}$, then there are $kappa_{1},kappa_{2}in Lip(Y,rho,eta)$ with $kappa_{1}kappa_{2}=1_{Y}$ and a Lipschitz homeomorphism $varphi$ from $(Y,rho)$ to $(X,d)$ with $tau circvarphi=varphi circ eta$ on $Y$ such that $T_{j}(f)=kappa_{j}cdot(S_{j}(f)circvarphi)$ for all $fin mathcal{A}$ and $j=1,2$. Moreover, we show that the same result holds for surjective mappings $S_{1},S_{2}:mathcal{A}rightarrow mathcal{A}$ and $T_{1},T_{2}:mathcal{A}rightarrow mathcal{B}$ that satisfy $sigma_{pi,mathcal{B}}(T_{1}(f)T_{2}(g))capsigma_{pi,mathcal{A}}(S_{1}(f)S_{2}(g))neqemptyset$ for all $f,gin mathcal{A}$.
In this paper, we consider a multi-species Lotka-Volterra type competitive system with delays and feedback controls on time scales. A general criteria on the permanence is established and then by constructing suitable Lyapunov functionals, sufficient conditions are derived for the existence anduniform asymptotic stability of unique positive almost periodic solution of the system.
